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Photon Momentum and Uncertainty Principle

Outline 

- Photons Have Momentum (Compton Scattering)

- Wavepackets - Review

- Resolution of an Optical Microscope

- Heisenbergs Uncertainty Principle

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TRUE / FALSE

1.  The photoelectric effect was used to show that light

was composed of packets of energy proportional toits frequency.

2.  The number of photons present in a beam of light is

simply the intensity  I  divided by the photon energy hν.

3. 

Infrared light at a wavelength of 1.24 microns hasphoton energy of 1.5 eV.

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 So is Light aWave or a Particle ? 

 

Light is always both

Wave and Particle !

On macroscopic scales, large number of photons looklike they exhibit only wave phenomena.

A single photon is still a wave, but your act of trying tomeasure it makes it look like a localized particle.

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Do Photons Have Momentum ? 

What is momentum ?

1E  =   mv2

  1 1= (mv)

2 2  · v =   p

2  · v

Photons have energy and a finite velocity so theremust be some momentum associated with photons !

E hν  p = =

c c

Just like Energy,TOTAL MOMENTUM IS ALWAYS CONSERVED

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 Photon Momentum

IN FREE SPACE:

E     ωE  = cp ⇒  p = = =   kc c

IN OPTICAL MATERIALS:

E     ωE  = v p p ⇒  p = = =   kvacnv p   v p

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φ

θλi

λf 

Compton Scattering

incidentphoton

targetelectron

at rest

recoilelectron

scatteredphoton

Compton found that if you treat the photons as if they were particlesE   = hc/λ   p = h/λof zero mass, with energy and momentum .

the collision behaves just as if it were two billiard balls colliding !(with total momentum always conserved)

In 1924, A. H. Compton performed an experimentwhere X-rays impinged on matter,

and he measured the scattered radiation.

It was found that the scatteredX-ray did not have the same

wavelength !

hλf  − λi  = ∆λ = (1moc   − cos θ)

Image by GFHund http://commons.wikimedia.org/wiki/File:Compton,Arthur_1929_Chicago.jpg

Wikimedia Commons.

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Manifestation of the Photon Momentum

Conservation of linearmomentum implies that an

atom recoils when it

undergoes spontaneous

emission. The direction of

 photon emission (and atomicrecoil) is not predictable.

 A w be

w be

 pcollimatingdiaphragms

beam spreads laterallybecause of spontaneous

emission s

ell-collimated atomicam of excited atoms

ill spread laterallycause of the recoil

associated with

ontaneous emission.

 A source emitting a sphericalwave cannot recoil, because

the spherical symmetry of

the wave prevents it from

carrying any linear

momentum from the source.

SOURCE EMITT ING A PHOTON

SOURCE EMIT T ING AN EM WAVE

excited atom

de -excited atom

 photonθ

source ofexcited atoms

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SOLAR SAILat rest

INCOMING PHOTONS1000 W/m2

every secondphotons with momentum

+ (1000 J/m2)/c impact the sail

SOLAR SAILmoves

REFLECTED PHOTONS1000 W/m2

every secondphotons with momentum

- (1000 J/m2)/c leave the sail

withmomentum

+ (2000 J/m2)/c

… and gets that

much more

Pressure acting on the sail = (2000 J/m2) /c /second = 6.7 Newtons/km2 momentumevery second …

Photon Momentum - Moves Solar SailsPhoton Momentum - Moves Solar Sails

Image by D. Kassing http://en.wikipedia.org/wiki/File:SolarSail-DLR-ESA.jpg on Wikipedia

Image in the Public Domain

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Optoelectric Tweezers

Transforms optical energy to electrical energy through theuse of a photoconductive surface. The idea is similar to

that used in the ubiquitous office copier machine. In

xerography, a document is scanned and transferred onto a

photosensitive drum, which attracts dyes of carbon

particles that are rolled onto a piece of paper to reproducethe image.

In this case, the researchers use a photosensitive surface

made of amorphous silicon, a common material used in

solar cells and flat-panel displays. Microscopic polystyrene

particles suspended in a liquid were sandwiched between apiece of glass and the photoconductive material. Wherever

light would hit the photosensitive material, it would

behave like a conducting electrode, while areas not

exposed to light would behave like a non-conductingtng

e

ITO Glass

ProjectedImage

Glass Substrate

Nitride

Undoped a-Si:H

n+a-Si:H

ITO

10x

LighEmitti

DiodDMD Microdisplay

insulator. Once a light source is removed, the

photosensitive material returns to normal.

Depending upon the properties of the particles or cellsbeing studied, they will either be attracted to or repelled

by the electric field generated by the optoelectronic

tweezer. Either way, the researchers can use that behavior

to scoot particles where they want them to go.

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Review of Wavepackets

WHAT WOULD WE GET IF WE SUPERIMPOSED

WAVES OF MANY DIFFERENT FREQUENCIES ?

+∞

 E  = E    (   kzo 

  f  (k) e+j ωt−   )dk−∞

LET’S SET THE FREQUENCY DISTRIBUTION as GAUSSIAN

1f  (k) = √    o

2πσk   kf (k)

k

σk

  (k −   2k   )

exp   −  2σ2

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Reminder: Gaussian Distributionσ50% of data within ±√ 

2

1   −   2

x   √   −

(x µ)

f  ( ) =   e   2  μ specifies the position of the bell curve’s central peak

2σ σ  specifies the half-distance between inflection points2πσ

FOURIER TRANSFORM OF A GAUSSIAN IS A GAUSSIAN

F x   e−ax

2

(k) = π

e−

  2 2π k

aa

0.4

0.3

0.2

0.1

µ0.1%   0.1%

2.1%   2.1%

13.6%13.6%

34.1% 34.1%

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REMEMBER:

Gaussian Wavepacket in Space

Re{E n (z, to)}

WE SET THE FREQUENCY DISTRIBUTION as GAUSSIAN

t = to

n

Re{E n (z, to)

} SUM OF SINUSOIDS= WAVEPACKET1 (k

f  (k) =  √    exp  − 2ko)

2πσ   2k

−

  2σk

f (k)

ko k

σk

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Gaussian W Gaussian W Gaussian Wavepacket in Spaceavepacket in Spaceavepacket in Space

k   2E (z, t) = E oexp

  σ2

−   (ct2

  − z)

cos(ωot− koz)

GAUSSIANENVELOPE

∆z = √ 2σk1

∆k∆z  = 1/2 

f (k)

ko k

∆k =  σk√ 

2

In free space …

ω   2πE k = =

c hc… this plot then shows the PROBABILITY OF

WHICH k (or ENERGY) EM WAVES are

MOST LIKELY TO BE IN THE WAVEPACKET

WAVE PACKET

λo

λo  = 2π

ko

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UNCERTAINTY RELATIONS

Gaussian Wavepacket in Time

σ2E (z, t) = E oexp   − k (ct

2  − 2z) cos (ωot− koz)

WAVE PACKETλo

λo =  2π

ko

∆z = ∆tn

nk = ∆ω

c

c

∆

∆k∆z = 1/2

∆ω∆t = 1/2

∆ p∆z  =   /2

∆E ∆t =  

/2

∆k

∆z

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 Crookes Radiometer 

Today   s Culture Moment 

The crookes radiometer spins because of

thermal processes, but he initially guessed itwas due to photon momentum…

invented in 1873 by the chemist

Sir William Crookes

Image by Rob Iretonhttp://www.flickr.com

/photos/aoisakana/

3308350714/ from Flickr.

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Heisenberg realized that ...

  In the world of very small particles, one cannot measure any

property of a particle without interacting with it in some way

  This introduces an unavoidable uncertainty into the result

  One can never measure all the

properties exactly

Werner Heisenberg (1901-1976)

Image in the Public Domain16

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t

BEFORE

ELECTRON-PHOTONCOLLISION

electron

cidenthoton

 Measuring Position and Momentum

of an Electron

  Shine light on electron and detect

reflected light using a microscope

  Minimum uncertainty in position

is given by the wavelength of the ligh

  So to determine the position

accurately, it is necessary to use

light with a short wavelengthin p

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on

AFTER

ELECTRON-PHOTONCOLLISION

recoilingelectron

red n

  By Plancks law E  = hc/λ, a photon with a

short wavelength has a large energy

  Thus, it would impart a large kick to the electr

  But to determine its momentum accurately,

electron must only be given a small kick

  This means using light of long wavelength !

Measuring Position and Momentumof an Electron

scatte photo

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 Implications

  It is impossible to know both the position and momentum

exactly, i.e., Δ x =0 and Δ p=0

  These uncertainties are inherent in the physical world and

have nothing to do with the skill of the observer

  Because h is so small, these uncertainties are not observable in

normal everyday situations

    .= 1 054 × 10−34 [J · s]

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Example of Baseball

  A pitcher throws a 0.1-kg baseball at 40 m/s

  So momentum is 0.1 x 40 = 4 kg m/s

 

Suppose the momentum is measured to an accuracyof 1 percent , i.e.,

Δp = 0.01  p = 4 x 10-2 kg m/s

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Example of Baseball (contd)

  The uncertainty in position is then

h∆x ≥   .= 1 3

4π∆ p  × 10−33m

  No wonder one does not observe the effects of the

uncertainty principle in everyday life!

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 Example of Electron

  Same situation, but baseball replaced by an electron

which has mass 9.11 x 10-31 kg traveling at 40 m/s

  So momentum = 3.6 x 10-29 kg m/s

and its uncertainty = 3.6 x 10-31

 kg m/s

  The uncertainty in position is then

h∆x ≥   .= 1 4

4π∆ p  × 10−4m

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 Classical World 

  The observer is objective and passive

  Physical events happen independently of whether there is a

observer or not

  This is known as objective reality

n

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Role of an Observer in

Quantum Mechanics

  The observer is not objective and passive

  The act of observation changes the physical system irrevocably

 

This is known as subjective reality

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One might ask:

If l i ght can behave l i ke a par t i cle,might par t ic les act l i ke waves  ?

YES !Particles, like photons, also have a wavelength given by:

The wavelength of a particle depends on its momentum,just like a photon!

The main difference is that matter particles have mass,and hotons dont!

λ = h/p = h/mv

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Matter Waves

Compute the wavelength of a 10 [g] bullet moving at 1000 [m/s].

λ = h/mv = 6.6x10-34 [J s] / (0.01 [kg])(1000 [m/s])

= 6.6x10-35 [m]

This is immeasureably small 

For ordinary everyday objects, 

we dont experience that

MATTER CAN BEHAVE AS A WAVE  

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GammaRays

X Rays

UV Rays

InfraredRadiation

Microwaves

But, what about small particles ? 

RadioWaves

Compute the wavelength of an electron(m = 9.1x10-31 [kg]) moving at 1x107 [m/s].

λ = h/mv

= 6.6x10-34 [J s]/(9.1x10-31 [kg])(1x107 [m/s]) = 7.3x10-11 [m].

= 0.073 [nm] 

These electronshave a wavelength in the region

of X-rays

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 Wavelength versus Size

With a visible light microscope, we are limited to being

able to resolve objects which are at least about

0.5*10-6 m = 0.5 μm = 500 nm in size.

This is because visible light, with a wavelength of ~500 nm cannotresolve objects whose size is smaller than its wavelength.

Image is in the public domain

Bacteria, as viewed

using visible light

Image is in the public domain

Bacteria, as viewed

using electrons!

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 Electron Microscope

This image was taken with a

Scanning Electron Microscope (SEM).

SEM can resolve features as small as 5 nm.

This is about 100 times better than can be

done with visible light microscopes!

The electron microscope is a device which uses the

wave behavior of electrons to make images

which are otherwise too small for visible light!

IMPORTANT POINT:

High energy particles can be used to reveal the structure of matter !

Image in the Public Domain

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SEM of various types of pollenImage in the Public Domain

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SEM of an ant headImage in the Public Domain

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 Summary 

  Light is made up of photons, but in macroscopic situations

it is often fine to treat it as a wave.

  Photons carry both energy & momentum.

E  =  hc/λ   p = E/c = h/λ

 

Matter also exhibits wave properties. For an object of mass m,and velocity, v, the object has a wavelength, λ = h / mv

  One can probe see the fine details of matter by using

high energy particles (they have a small wavelength !)

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6.007 Electromagnetic Energy: From Motors to Lasers

Spring 2011

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